Properties

Label 912.2.a.m.1.2
Level $912$
Weight $2$
Character 912.1
Self dual yes
Analytic conductor $7.282$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.28235666434\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 912.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.56155 q^{5} -2.56155 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.56155 q^{5} -2.56155 q^{7} +1.00000 q^{9} -1.43845 q^{11} -5.12311 q^{13} -2.56155 q^{15} -5.68466 q^{17} -1.00000 q^{19} +2.56155 q^{21} -0.876894 q^{23} +1.56155 q^{25} -1.00000 q^{27} +8.24621 q^{29} +2.00000 q^{31} +1.43845 q^{33} -6.56155 q^{35} -8.00000 q^{37} +5.12311 q^{39} +3.12311 q^{41} -2.56155 q^{43} +2.56155 q^{45} -5.68466 q^{47} -0.438447 q^{49} +5.68466 q^{51} -12.2462 q^{53} -3.68466 q^{55} +1.00000 q^{57} -12.0000 q^{59} -5.68466 q^{61} -2.56155 q^{63} -13.1231 q^{65} +10.2462 q^{67} +0.876894 q^{69} +11.9309 q^{73} -1.56155 q^{75} +3.68466 q^{77} +13.3693 q^{79} +1.00000 q^{81} -4.00000 q^{83} -14.5616 q^{85} -8.24621 q^{87} -6.00000 q^{89} +13.1231 q^{91} -2.00000 q^{93} -2.56155 q^{95} -12.2462 q^{97} -1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 2 q^{13} - q^{15} + q^{17} - 2 q^{19} + q^{21} - 10 q^{23} - q^{25} - 2 q^{27} + 4 q^{31} + 7 q^{33} - 9 q^{35} - 16 q^{37} + 2 q^{39} - 2 q^{41} - q^{43} + q^{45} + q^{47} - 5 q^{49} - q^{51} - 8 q^{53} + 5 q^{55} + 2 q^{57} - 24 q^{59} + q^{61} - q^{63} - 18 q^{65} + 4 q^{67} + 10 q^{69} - 5 q^{73} + q^{75} - 5 q^{77} + 2 q^{79} + 2 q^{81} - 8 q^{83} - 25 q^{85} - 12 q^{89} + 18 q^{91} - 4 q^{93} - q^{95} - 8 q^{97} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.56155 1.14556 0.572781 0.819709i \(-0.305865\pi\)
0.572781 + 0.819709i \(0.305865\pi\)
\(6\) 0 0
\(7\) −2.56155 −0.968176 −0.484088 0.875019i \(-0.660849\pi\)
−0.484088 + 0.875019i \(0.660849\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.43845 −0.433708 −0.216854 0.976204i \(-0.569580\pi\)
−0.216854 + 0.976204i \(0.569580\pi\)
\(12\) 0 0
\(13\) −5.12311 −1.42089 −0.710447 0.703751i \(-0.751507\pi\)
−0.710447 + 0.703751i \(0.751507\pi\)
\(14\) 0 0
\(15\) −2.56155 −0.661390
\(16\) 0 0
\(17\) −5.68466 −1.37873 −0.689366 0.724413i \(-0.742111\pi\)
−0.689366 + 0.724413i \(0.742111\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 2.56155 0.558977
\(22\) 0 0
\(23\) −0.876894 −0.182845 −0.0914226 0.995812i \(-0.529141\pi\)
−0.0914226 + 0.995812i \(0.529141\pi\)
\(24\) 0 0
\(25\) 1.56155 0.312311
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.24621 1.53128 0.765641 0.643268i \(-0.222422\pi\)
0.765641 + 0.643268i \(0.222422\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 1.43845 0.250402
\(34\) 0 0
\(35\) −6.56155 −1.10910
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 5.12311 0.820353
\(40\) 0 0
\(41\) 3.12311 0.487747 0.243874 0.969807i \(-0.421582\pi\)
0.243874 + 0.969807i \(0.421582\pi\)
\(42\) 0 0
\(43\) −2.56155 −0.390633 −0.195317 0.980740i \(-0.562573\pi\)
−0.195317 + 0.980740i \(0.562573\pi\)
\(44\) 0 0
\(45\) 2.56155 0.381854
\(46\) 0 0
\(47\) −5.68466 −0.829193 −0.414596 0.910005i \(-0.636077\pi\)
−0.414596 + 0.910005i \(0.636077\pi\)
\(48\) 0 0
\(49\) −0.438447 −0.0626353
\(50\) 0 0
\(51\) 5.68466 0.796011
\(52\) 0 0
\(53\) −12.2462 −1.68215 −0.841073 0.540921i \(-0.818076\pi\)
−0.841073 + 0.540921i \(0.818076\pi\)
\(54\) 0 0
\(55\) −3.68466 −0.496839
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −5.68466 −0.727846 −0.363923 0.931429i \(-0.618563\pi\)
−0.363923 + 0.931429i \(0.618563\pi\)
\(62\) 0 0
\(63\) −2.56155 −0.322725
\(64\) 0 0
\(65\) −13.1231 −1.62772
\(66\) 0 0
\(67\) 10.2462 1.25177 0.625887 0.779914i \(-0.284737\pi\)
0.625887 + 0.779914i \(0.284737\pi\)
\(68\) 0 0
\(69\) 0.876894 0.105566
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 11.9309 1.39640 0.698201 0.715901i \(-0.253984\pi\)
0.698201 + 0.715901i \(0.253984\pi\)
\(74\) 0 0
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) 3.68466 0.419906
\(78\) 0 0
\(79\) 13.3693 1.50417 0.752083 0.659069i \(-0.229049\pi\)
0.752083 + 0.659069i \(0.229049\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −14.5616 −1.57942
\(86\) 0 0
\(87\) −8.24621 −0.884087
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 13.1231 1.37568
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) −2.56155 −0.262810
\(96\) 0 0
\(97\) −12.2462 −1.24341 −0.621707 0.783250i \(-0.713561\pi\)
−0.621707 + 0.783250i \(0.713561\pi\)
\(98\) 0 0
\(99\) −1.43845 −0.144569
\(100\) 0 0
\(101\) −1.12311 −0.111753 −0.0558766 0.998438i \(-0.517795\pi\)
−0.0558766 + 0.998438i \(0.517795\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 6.56155 0.640342
\(106\) 0 0
\(107\) −11.3693 −1.09911 −0.549557 0.835456i \(-0.685203\pi\)
−0.549557 + 0.835456i \(0.685203\pi\)
\(108\) 0 0
\(109\) 14.2462 1.36454 0.682270 0.731101i \(-0.260993\pi\)
0.682270 + 0.731101i \(0.260993\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −2.24621 −0.209460
\(116\) 0 0
\(117\) −5.12311 −0.473631
\(118\) 0 0
\(119\) 14.5616 1.33486
\(120\) 0 0
\(121\) −8.93087 −0.811897
\(122\) 0 0
\(123\) −3.12311 −0.281601
\(124\) 0 0
\(125\) −8.80776 −0.787790
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 0 0
\(129\) 2.56155 0.225532
\(130\) 0 0
\(131\) −8.31534 −0.726515 −0.363257 0.931689i \(-0.618335\pi\)
−0.363257 + 0.931689i \(0.618335\pi\)
\(132\) 0 0
\(133\) 2.56155 0.222115
\(134\) 0 0
\(135\) −2.56155 −0.220463
\(136\) 0 0
\(137\) 6.80776 0.581627 0.290813 0.956780i \(-0.406074\pi\)
0.290813 + 0.956780i \(0.406074\pi\)
\(138\) 0 0
\(139\) −12.8078 −1.08634 −0.543170 0.839623i \(-0.682776\pi\)
−0.543170 + 0.839623i \(0.682776\pi\)
\(140\) 0 0
\(141\) 5.68466 0.478735
\(142\) 0 0
\(143\) 7.36932 0.616253
\(144\) 0 0
\(145\) 21.1231 1.75418
\(146\) 0 0
\(147\) 0.438447 0.0361625
\(148\) 0 0
\(149\) −16.8078 −1.37695 −0.688473 0.725262i \(-0.741719\pi\)
−0.688473 + 0.725262i \(0.741719\pi\)
\(150\) 0 0
\(151\) 8.87689 0.722391 0.361196 0.932490i \(-0.382369\pi\)
0.361196 + 0.932490i \(0.382369\pi\)
\(152\) 0 0
\(153\) −5.68466 −0.459577
\(154\) 0 0
\(155\) 5.12311 0.411498
\(156\) 0 0
\(157\) 4.24621 0.338885 0.169442 0.985540i \(-0.445803\pi\)
0.169442 + 0.985540i \(0.445803\pi\)
\(158\) 0 0
\(159\) 12.2462 0.971188
\(160\) 0 0
\(161\) 2.24621 0.177026
\(162\) 0 0
\(163\) −14.2462 −1.11585 −0.557925 0.829892i \(-0.688402\pi\)
−0.557925 + 0.829892i \(0.688402\pi\)
\(164\) 0 0
\(165\) 3.68466 0.286850
\(166\) 0 0
\(167\) 17.1231 1.32503 0.662513 0.749051i \(-0.269490\pi\)
0.662513 + 0.749051i \(0.269490\pi\)
\(168\) 0 0
\(169\) 13.2462 1.01894
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 12.2462 0.931062 0.465531 0.885032i \(-0.345863\pi\)
0.465531 + 0.885032i \(0.345863\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 6.87689 0.514003 0.257002 0.966411i \(-0.417265\pi\)
0.257002 + 0.966411i \(0.417265\pi\)
\(180\) 0 0
\(181\) 6.87689 0.511156 0.255578 0.966789i \(-0.417734\pi\)
0.255578 + 0.966789i \(0.417734\pi\)
\(182\) 0 0
\(183\) 5.68466 0.420222
\(184\) 0 0
\(185\) −20.4924 −1.50663
\(186\) 0 0
\(187\) 8.17708 0.597967
\(188\) 0 0
\(189\) 2.56155 0.186326
\(190\) 0 0
\(191\) 13.6847 0.990187 0.495094 0.868840i \(-0.335134\pi\)
0.495094 + 0.868840i \(0.335134\pi\)
\(192\) 0 0
\(193\) 27.1231 1.95236 0.976182 0.216954i \(-0.0696120\pi\)
0.976182 + 0.216954i \(0.0696120\pi\)
\(194\) 0 0
\(195\) 13.1231 0.939765
\(196\) 0 0
\(197\) 23.3693 1.66499 0.832497 0.554029i \(-0.186910\pi\)
0.832497 + 0.554029i \(0.186910\pi\)
\(198\) 0 0
\(199\) 25.9309 1.83819 0.919095 0.394035i \(-0.128921\pi\)
0.919095 + 0.394035i \(0.128921\pi\)
\(200\) 0 0
\(201\) −10.2462 −0.722712
\(202\) 0 0
\(203\) −21.1231 −1.48255
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 0 0
\(207\) −0.876894 −0.0609484
\(208\) 0 0
\(209\) 1.43845 0.0994995
\(210\) 0 0
\(211\) −22.2462 −1.53149 −0.765746 0.643143i \(-0.777630\pi\)
−0.765746 + 0.643143i \(0.777630\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.56155 −0.447494
\(216\) 0 0
\(217\) −5.12311 −0.347779
\(218\) 0 0
\(219\) −11.9309 −0.806213
\(220\) 0 0
\(221\) 29.1231 1.95903
\(222\) 0 0
\(223\) −13.3693 −0.895276 −0.447638 0.894215i \(-0.647735\pi\)
−0.447638 + 0.894215i \(0.647735\pi\)
\(224\) 0 0
\(225\) 1.56155 0.104104
\(226\) 0 0
\(227\) 17.1231 1.13650 0.568250 0.822856i \(-0.307621\pi\)
0.568250 + 0.822856i \(0.307621\pi\)
\(228\) 0 0
\(229\) −0.561553 −0.0371085 −0.0185542 0.999828i \(-0.505906\pi\)
−0.0185542 + 0.999828i \(0.505906\pi\)
\(230\) 0 0
\(231\) −3.68466 −0.242433
\(232\) 0 0
\(233\) −2.31534 −0.151683 −0.0758415 0.997120i \(-0.524164\pi\)
−0.0758415 + 0.997120i \(0.524164\pi\)
\(234\) 0 0
\(235\) −14.5616 −0.949891
\(236\) 0 0
\(237\) −13.3693 −0.868430
\(238\) 0 0
\(239\) −26.8078 −1.73405 −0.867025 0.498265i \(-0.833971\pi\)
−0.867025 + 0.498265i \(0.833971\pi\)
\(240\) 0 0
\(241\) 13.3693 0.861193 0.430597 0.902544i \(-0.358303\pi\)
0.430597 + 0.902544i \(0.358303\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.12311 −0.0717526
\(246\) 0 0
\(247\) 5.12311 0.325975
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −12.8078 −0.808419 −0.404209 0.914666i \(-0.632453\pi\)
−0.404209 + 0.914666i \(0.632453\pi\)
\(252\) 0 0
\(253\) 1.26137 0.0793014
\(254\) 0 0
\(255\) 14.5616 0.911880
\(256\) 0 0
\(257\) 4.87689 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(258\) 0 0
\(259\) 20.4924 1.27334
\(260\) 0 0
\(261\) 8.24621 0.510428
\(262\) 0 0
\(263\) 2.31534 0.142770 0.0713850 0.997449i \(-0.477258\pi\)
0.0713850 + 0.997449i \(0.477258\pi\)
\(264\) 0 0
\(265\) −31.3693 −1.92700
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) −13.1231 −0.794246
\(274\) 0 0
\(275\) −2.24621 −0.135452
\(276\) 0 0
\(277\) −6.31534 −0.379452 −0.189726 0.981837i \(-0.560760\pi\)
−0.189726 + 0.981837i \(0.560760\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −7.75379 −0.462552 −0.231276 0.972888i \(-0.574290\pi\)
−0.231276 + 0.972888i \(0.574290\pi\)
\(282\) 0 0
\(283\) −16.3153 −0.969846 −0.484923 0.874557i \(-0.661152\pi\)
−0.484923 + 0.874557i \(0.661152\pi\)
\(284\) 0 0
\(285\) 2.56155 0.151733
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) 15.3153 0.900902
\(290\) 0 0
\(291\) 12.2462 0.717886
\(292\) 0 0
\(293\) 7.12311 0.416136 0.208068 0.978114i \(-0.433282\pi\)
0.208068 + 0.978114i \(0.433282\pi\)
\(294\) 0 0
\(295\) −30.7386 −1.78967
\(296\) 0 0
\(297\) 1.43845 0.0834672
\(298\) 0 0
\(299\) 4.49242 0.259804
\(300\) 0 0
\(301\) 6.56155 0.378202
\(302\) 0 0
\(303\) 1.12311 0.0645207
\(304\) 0 0
\(305\) −14.5616 −0.833792
\(306\) 0 0
\(307\) −28.4924 −1.62615 −0.813074 0.582160i \(-0.802208\pi\)
−0.813074 + 0.582160i \(0.802208\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) −7.93087 −0.449718 −0.224859 0.974391i \(-0.572192\pi\)
−0.224859 + 0.974391i \(0.572192\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) −6.56155 −0.369702
\(316\) 0 0
\(317\) −27.1231 −1.52339 −0.761693 0.647938i \(-0.775631\pi\)
−0.761693 + 0.647938i \(0.775631\pi\)
\(318\) 0 0
\(319\) −11.8617 −0.664130
\(320\) 0 0
\(321\) 11.3693 0.634573
\(322\) 0 0
\(323\) 5.68466 0.316303
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) −14.2462 −0.787817
\(328\) 0 0
\(329\) 14.5616 0.802804
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) 26.2462 1.43398
\(336\) 0 0
\(337\) −28.7386 −1.56549 −0.782747 0.622341i \(-0.786182\pi\)
−0.782747 + 0.622341i \(0.786182\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) −2.87689 −0.155793
\(342\) 0 0
\(343\) 19.0540 1.02882
\(344\) 0 0
\(345\) 2.24621 0.120932
\(346\) 0 0
\(347\) 34.4233 1.84794 0.923970 0.382466i \(-0.124925\pi\)
0.923970 + 0.382466i \(0.124925\pi\)
\(348\) 0 0
\(349\) 29.6847 1.58898 0.794492 0.607275i \(-0.207737\pi\)
0.794492 + 0.607275i \(0.207737\pi\)
\(350\) 0 0
\(351\) 5.12311 0.273451
\(352\) 0 0
\(353\) −11.7538 −0.625591 −0.312796 0.949820i \(-0.601265\pi\)
−0.312796 + 0.949820i \(0.601265\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −14.5616 −0.770679
\(358\) 0 0
\(359\) −4.56155 −0.240750 −0.120375 0.992729i \(-0.538410\pi\)
−0.120375 + 0.992729i \(0.538410\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 8.93087 0.468749
\(364\) 0 0
\(365\) 30.5616 1.59966
\(366\) 0 0
\(367\) 14.2462 0.743646 0.371823 0.928304i \(-0.378733\pi\)
0.371823 + 0.928304i \(0.378733\pi\)
\(368\) 0 0
\(369\) 3.12311 0.162582
\(370\) 0 0
\(371\) 31.3693 1.62861
\(372\) 0 0
\(373\) −3.50758 −0.181615 −0.0908077 0.995868i \(-0.528945\pi\)
−0.0908077 + 0.995868i \(0.528945\pi\)
\(374\) 0 0
\(375\) 8.80776 0.454831
\(376\) 0 0
\(377\) −42.2462 −2.17579
\(378\) 0 0
\(379\) 29.1231 1.49595 0.747977 0.663725i \(-0.231025\pi\)
0.747977 + 0.663725i \(0.231025\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) −21.1231 −1.07934 −0.539670 0.841877i \(-0.681451\pi\)
−0.539670 + 0.841877i \(0.681451\pi\)
\(384\) 0 0
\(385\) 9.43845 0.481028
\(386\) 0 0
\(387\) −2.56155 −0.130211
\(388\) 0 0
\(389\) −9.93087 −0.503515 −0.251758 0.967790i \(-0.581009\pi\)
−0.251758 + 0.967790i \(0.581009\pi\)
\(390\) 0 0
\(391\) 4.98485 0.252094
\(392\) 0 0
\(393\) 8.31534 0.419453
\(394\) 0 0
\(395\) 34.2462 1.72311
\(396\) 0 0
\(397\) −10.3153 −0.517712 −0.258856 0.965916i \(-0.583346\pi\)
−0.258856 + 0.965916i \(0.583346\pi\)
\(398\) 0 0
\(399\) −2.56155 −0.128238
\(400\) 0 0
\(401\) −25.3693 −1.26688 −0.633442 0.773790i \(-0.718358\pi\)
−0.633442 + 0.773790i \(0.718358\pi\)
\(402\) 0 0
\(403\) −10.2462 −0.510400
\(404\) 0 0
\(405\) 2.56155 0.127285
\(406\) 0 0
\(407\) 11.5076 0.570409
\(408\) 0 0
\(409\) 16.2462 0.803323 0.401662 0.915788i \(-0.368433\pi\)
0.401662 + 0.915788i \(0.368433\pi\)
\(410\) 0 0
\(411\) −6.80776 −0.335802
\(412\) 0 0
\(413\) 30.7386 1.51255
\(414\) 0 0
\(415\) −10.2462 −0.502967
\(416\) 0 0
\(417\) 12.8078 0.627199
\(418\) 0 0
\(419\) −22.7386 −1.11085 −0.555427 0.831565i \(-0.687445\pi\)
−0.555427 + 0.831565i \(0.687445\pi\)
\(420\) 0 0
\(421\) 13.1231 0.639581 0.319791 0.947488i \(-0.396387\pi\)
0.319791 + 0.947488i \(0.396387\pi\)
\(422\) 0 0
\(423\) −5.68466 −0.276398
\(424\) 0 0
\(425\) −8.87689 −0.430593
\(426\) 0 0
\(427\) 14.5616 0.704683
\(428\) 0 0
\(429\) −7.36932 −0.355794
\(430\) 0 0
\(431\) −29.6155 −1.42653 −0.713265 0.700895i \(-0.752784\pi\)
−0.713265 + 0.700895i \(0.752784\pi\)
\(432\) 0 0
\(433\) 8.24621 0.396288 0.198144 0.980173i \(-0.436509\pi\)
0.198144 + 0.980173i \(0.436509\pi\)
\(434\) 0 0
\(435\) −21.1231 −1.01278
\(436\) 0 0
\(437\) 0.876894 0.0419475
\(438\) 0 0
\(439\) −11.7538 −0.560978 −0.280489 0.959857i \(-0.590497\pi\)
−0.280489 + 0.959857i \(0.590497\pi\)
\(440\) 0 0
\(441\) −0.438447 −0.0208784
\(442\) 0 0
\(443\) −32.8078 −1.55874 −0.779372 0.626562i \(-0.784462\pi\)
−0.779372 + 0.626562i \(0.784462\pi\)
\(444\) 0 0
\(445\) −15.3693 −0.728575
\(446\) 0 0
\(447\) 16.8078 0.794980
\(448\) 0 0
\(449\) 10.6307 0.501693 0.250846 0.968027i \(-0.419291\pi\)
0.250846 + 0.968027i \(0.419291\pi\)
\(450\) 0 0
\(451\) −4.49242 −0.211540
\(452\) 0 0
\(453\) −8.87689 −0.417073
\(454\) 0 0
\(455\) 33.6155 1.57592
\(456\) 0 0
\(457\) −2.94602 −0.137809 −0.0689046 0.997623i \(-0.521950\pi\)
−0.0689046 + 0.997623i \(0.521950\pi\)
\(458\) 0 0
\(459\) 5.68466 0.265337
\(460\) 0 0
\(461\) −21.3002 −0.992049 −0.496024 0.868309i \(-0.665207\pi\)
−0.496024 + 0.868309i \(0.665207\pi\)
\(462\) 0 0
\(463\) −11.1922 −0.520147 −0.260074 0.965589i \(-0.583747\pi\)
−0.260074 + 0.965589i \(0.583747\pi\)
\(464\) 0 0
\(465\) −5.12311 −0.237578
\(466\) 0 0
\(467\) 17.9309 0.829742 0.414871 0.909880i \(-0.363827\pi\)
0.414871 + 0.909880i \(0.363827\pi\)
\(468\) 0 0
\(469\) −26.2462 −1.21194
\(470\) 0 0
\(471\) −4.24621 −0.195655
\(472\) 0 0
\(473\) 3.68466 0.169421
\(474\) 0 0
\(475\) −1.56155 −0.0716490
\(476\) 0 0
\(477\) −12.2462 −0.560715
\(478\) 0 0
\(479\) 9.36932 0.428095 0.214048 0.976823i \(-0.431335\pi\)
0.214048 + 0.976823i \(0.431335\pi\)
\(480\) 0 0
\(481\) 40.9848 1.86875
\(482\) 0 0
\(483\) −2.24621 −0.102206
\(484\) 0 0
\(485\) −31.3693 −1.42441
\(486\) 0 0
\(487\) 24.8769 1.12728 0.563640 0.826021i \(-0.309401\pi\)
0.563640 + 0.826021i \(0.309401\pi\)
\(488\) 0 0
\(489\) 14.2462 0.644236
\(490\) 0 0
\(491\) −34.7386 −1.56773 −0.783866 0.620930i \(-0.786755\pi\)
−0.783866 + 0.620930i \(0.786755\pi\)
\(492\) 0 0
\(493\) −46.8769 −2.11123
\(494\) 0 0
\(495\) −3.68466 −0.165613
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −36.1771 −1.61951 −0.809754 0.586769i \(-0.800400\pi\)
−0.809754 + 0.586769i \(0.800400\pi\)
\(500\) 0 0
\(501\) −17.1231 −0.765004
\(502\) 0 0
\(503\) 15.6155 0.696262 0.348131 0.937446i \(-0.386816\pi\)
0.348131 + 0.937446i \(0.386816\pi\)
\(504\) 0 0
\(505\) −2.87689 −0.128020
\(506\) 0 0
\(507\) −13.2462 −0.588285
\(508\) 0 0
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) −30.5616 −1.35196
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 15.3693 0.677253
\(516\) 0 0
\(517\) 8.17708 0.359628
\(518\) 0 0
\(519\) −12.2462 −0.537549
\(520\) 0 0
\(521\) −31.6155 −1.38510 −0.692551 0.721369i \(-0.743513\pi\)
−0.692551 + 0.721369i \(0.743513\pi\)
\(522\) 0 0
\(523\) 7.36932 0.322238 0.161119 0.986935i \(-0.448490\pi\)
0.161119 + 0.986935i \(0.448490\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) −11.3693 −0.495255
\(528\) 0 0
\(529\) −22.2311 −0.966568
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −16.0000 −0.693037
\(534\) 0 0
\(535\) −29.1231 −1.25910
\(536\) 0 0
\(537\) −6.87689 −0.296760
\(538\) 0 0
\(539\) 0.630683 0.0271654
\(540\) 0 0
\(541\) 2.80776 0.120715 0.0603576 0.998177i \(-0.480776\pi\)
0.0603576 + 0.998177i \(0.480776\pi\)
\(542\) 0 0
\(543\) −6.87689 −0.295116
\(544\) 0 0
\(545\) 36.4924 1.56316
\(546\) 0 0
\(547\) 41.6155 1.77935 0.889676 0.456593i \(-0.150930\pi\)
0.889676 + 0.456593i \(0.150930\pi\)
\(548\) 0 0
\(549\) −5.68466 −0.242615
\(550\) 0 0
\(551\) −8.24621 −0.351300
\(552\) 0 0
\(553\) −34.2462 −1.45630
\(554\) 0 0
\(555\) 20.4924 0.869855
\(556\) 0 0
\(557\) 41.4384 1.75580 0.877902 0.478841i \(-0.158943\pi\)
0.877902 + 0.478841i \(0.158943\pi\)
\(558\) 0 0
\(559\) 13.1231 0.555048
\(560\) 0 0
\(561\) −8.17708 −0.345237
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −5.12311 −0.215531
\(566\) 0 0
\(567\) −2.56155 −0.107575
\(568\) 0 0
\(569\) 14.4924 0.607554 0.303777 0.952743i \(-0.401752\pi\)
0.303777 + 0.952743i \(0.401752\pi\)
\(570\) 0 0
\(571\) 42.7386 1.78856 0.894278 0.447512i \(-0.147690\pi\)
0.894278 + 0.447512i \(0.147690\pi\)
\(572\) 0 0
\(573\) −13.6847 −0.571685
\(574\) 0 0
\(575\) −1.36932 −0.0571045
\(576\) 0 0
\(577\) −7.43845 −0.309667 −0.154833 0.987941i \(-0.549484\pi\)
−0.154833 + 0.987941i \(0.549484\pi\)
\(578\) 0 0
\(579\) −27.1231 −1.12720
\(580\) 0 0
\(581\) 10.2462 0.425084
\(582\) 0 0
\(583\) 17.6155 0.729561
\(584\) 0 0
\(585\) −13.1231 −0.542574
\(586\) 0 0
\(587\) −19.0540 −0.786442 −0.393221 0.919444i \(-0.628639\pi\)
−0.393221 + 0.919444i \(0.628639\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) −23.3693 −0.961285
\(592\) 0 0
\(593\) 5.50758 0.226169 0.113085 0.993585i \(-0.463927\pi\)
0.113085 + 0.993585i \(0.463927\pi\)
\(594\) 0 0
\(595\) 37.3002 1.52916
\(596\) 0 0
\(597\) −25.9309 −1.06128
\(598\) 0 0
\(599\) −48.4924 −1.98135 −0.990673 0.136258i \(-0.956492\pi\)
−0.990673 + 0.136258i \(0.956492\pi\)
\(600\) 0 0
\(601\) −17.8617 −0.728596 −0.364298 0.931283i \(-0.618691\pi\)
−0.364298 + 0.931283i \(0.618691\pi\)
\(602\) 0 0
\(603\) 10.2462 0.417258
\(604\) 0 0
\(605\) −22.8769 −0.930078
\(606\) 0 0
\(607\) −42.9848 −1.74470 −0.872351 0.488881i \(-0.837405\pi\)
−0.872351 + 0.488881i \(0.837405\pi\)
\(608\) 0 0
\(609\) 21.1231 0.855951
\(610\) 0 0
\(611\) 29.1231 1.17819
\(612\) 0 0
\(613\) −47.7926 −1.93033 −0.965163 0.261651i \(-0.915733\pi\)
−0.965163 + 0.261651i \(0.915733\pi\)
\(614\) 0 0
\(615\) −8.00000 −0.322591
\(616\) 0 0
\(617\) 39.3002 1.58217 0.791083 0.611709i \(-0.209518\pi\)
0.791083 + 0.611709i \(0.209518\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0.876894 0.0351886
\(622\) 0 0
\(623\) 15.3693 0.615759
\(624\) 0 0
\(625\) −30.3693 −1.21477
\(626\) 0 0
\(627\) −1.43845 −0.0574460
\(628\) 0 0
\(629\) 45.4773 1.81330
\(630\) 0 0
\(631\) 5.93087 0.236104 0.118052 0.993007i \(-0.462335\pi\)
0.118052 + 0.993007i \(0.462335\pi\)
\(632\) 0 0
\(633\) 22.2462 0.884208
\(634\) 0 0
\(635\) 46.1080 1.82974
\(636\) 0 0
\(637\) 2.24621 0.0889981
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.73863 0.345155 0.172578 0.984996i \(-0.444790\pi\)
0.172578 + 0.984996i \(0.444790\pi\)
\(642\) 0 0
\(643\) −47.0540 −1.85563 −0.927814 0.373044i \(-0.878314\pi\)
−0.927814 + 0.373044i \(0.878314\pi\)
\(644\) 0 0
\(645\) 6.56155 0.258361
\(646\) 0 0
\(647\) 1.19224 0.0468716 0.0234358 0.999725i \(-0.492539\pi\)
0.0234358 + 0.999725i \(0.492539\pi\)
\(648\) 0 0
\(649\) 17.2614 0.677568
\(650\) 0 0
\(651\) 5.12311 0.200790
\(652\) 0 0
\(653\) −13.3002 −0.520477 −0.260238 0.965544i \(-0.583801\pi\)
−0.260238 + 0.965544i \(0.583801\pi\)
\(654\) 0 0
\(655\) −21.3002 −0.832267
\(656\) 0 0
\(657\) 11.9309 0.465468
\(658\) 0 0
\(659\) 3.36932 0.131250 0.0656250 0.997844i \(-0.479096\pi\)
0.0656250 + 0.997844i \(0.479096\pi\)
\(660\) 0 0
\(661\) 6.73863 0.262102 0.131051 0.991376i \(-0.458165\pi\)
0.131051 + 0.991376i \(0.458165\pi\)
\(662\) 0 0
\(663\) −29.1231 −1.13105
\(664\) 0 0
\(665\) 6.56155 0.254446
\(666\) 0 0
\(667\) −7.23106 −0.279988
\(668\) 0 0
\(669\) 13.3693 0.516888
\(670\) 0 0
\(671\) 8.17708 0.315673
\(672\) 0 0
\(673\) 25.8617 0.996897 0.498448 0.866919i \(-0.333903\pi\)
0.498448 + 0.866919i \(0.333903\pi\)
\(674\) 0 0
\(675\) −1.56155 −0.0601042
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) 31.3693 1.20384
\(680\) 0 0
\(681\) −17.1231 −0.656159
\(682\) 0 0
\(683\) −47.8617 −1.83138 −0.915689 0.401887i \(-0.868354\pi\)
−0.915689 + 0.401887i \(0.868354\pi\)
\(684\) 0 0
\(685\) 17.4384 0.666289
\(686\) 0 0
\(687\) 0.561553 0.0214246
\(688\) 0 0
\(689\) 62.7386 2.39015
\(690\) 0 0
\(691\) −49.9309 −1.89946 −0.949730 0.313070i \(-0.898642\pi\)
−0.949730 + 0.313070i \(0.898642\pi\)
\(692\) 0 0
\(693\) 3.68466 0.139969
\(694\) 0 0
\(695\) −32.8078 −1.24447
\(696\) 0 0
\(697\) −17.7538 −0.672473
\(698\) 0 0
\(699\) 2.31534 0.0875743
\(700\) 0 0
\(701\) 14.8769 0.561893 0.280946 0.959723i \(-0.409352\pi\)
0.280946 + 0.959723i \(0.409352\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 14.5616 0.548420
\(706\) 0 0
\(707\) 2.87689 0.108197
\(708\) 0 0
\(709\) 42.4924 1.59584 0.797918 0.602766i \(-0.205935\pi\)
0.797918 + 0.602766i \(0.205935\pi\)
\(710\) 0 0
\(711\) 13.3693 0.501389
\(712\) 0 0
\(713\) −1.75379 −0.0656799
\(714\) 0 0
\(715\) 18.8769 0.705956
\(716\) 0 0
\(717\) 26.8078 1.00115
\(718\) 0 0
\(719\) 24.4233 0.910835 0.455418 0.890278i \(-0.349490\pi\)
0.455418 + 0.890278i \(0.349490\pi\)
\(720\) 0 0
\(721\) −15.3693 −0.572383
\(722\) 0 0
\(723\) −13.3693 −0.497210
\(724\) 0 0
\(725\) 12.8769 0.478236
\(726\) 0 0
\(727\) −9.43845 −0.350053 −0.175026 0.984564i \(-0.556001\pi\)
−0.175026 + 0.984564i \(0.556001\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 14.5616 0.538578
\(732\) 0 0
\(733\) 14.4924 0.535290 0.267645 0.963518i \(-0.413755\pi\)
0.267645 + 0.963518i \(0.413755\pi\)
\(734\) 0 0
\(735\) 1.12311 0.0414264
\(736\) 0 0
\(737\) −14.7386 −0.542905
\(738\) 0 0
\(739\) 20.1771 0.742226 0.371113 0.928588i \(-0.378976\pi\)
0.371113 + 0.928588i \(0.378976\pi\)
\(740\) 0 0
\(741\) −5.12311 −0.188202
\(742\) 0 0
\(743\) −7.50758 −0.275426 −0.137713 0.990472i \(-0.543975\pi\)
−0.137713 + 0.990472i \(0.543975\pi\)
\(744\) 0 0
\(745\) −43.0540 −1.57738
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 29.1231 1.06414
\(750\) 0 0
\(751\) 7.61553 0.277895 0.138947 0.990300i \(-0.455628\pi\)
0.138947 + 0.990300i \(0.455628\pi\)
\(752\) 0 0
\(753\) 12.8078 0.466741
\(754\) 0 0
\(755\) 22.7386 0.827544
\(756\) 0 0
\(757\) 5.05398 0.183690 0.0918449 0.995773i \(-0.470724\pi\)
0.0918449 + 0.995773i \(0.470724\pi\)
\(758\) 0 0
\(759\) −1.26137 −0.0457847
\(760\) 0 0
\(761\) −30.6695 −1.11177 −0.555884 0.831260i \(-0.687620\pi\)
−0.555884 + 0.831260i \(0.687620\pi\)
\(762\) 0 0
\(763\) −36.4924 −1.32111
\(764\) 0 0
\(765\) −14.5616 −0.526474
\(766\) 0 0
\(767\) 61.4773 2.21982
\(768\) 0 0
\(769\) 19.3002 0.695983 0.347991 0.937498i \(-0.386864\pi\)
0.347991 + 0.937498i \(0.386864\pi\)
\(770\) 0 0
\(771\) −4.87689 −0.175637
\(772\) 0 0
\(773\) −23.6155 −0.849392 −0.424696 0.905336i \(-0.639619\pi\)
−0.424696 + 0.905336i \(0.639619\pi\)
\(774\) 0 0
\(775\) 3.12311 0.112185
\(776\) 0 0
\(777\) −20.4924 −0.735162
\(778\) 0 0
\(779\) −3.12311 −0.111897
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −8.24621 −0.294696
\(784\) 0 0
\(785\) 10.8769 0.388213
\(786\) 0 0
\(787\) 6.24621 0.222653 0.111327 0.993784i \(-0.464490\pi\)
0.111327 + 0.993784i \(0.464490\pi\)
\(788\) 0 0
\(789\) −2.31534 −0.0824283
\(790\) 0 0
\(791\) 5.12311 0.182157
\(792\) 0 0
\(793\) 29.1231 1.03419
\(794\) 0 0
\(795\) 31.3693 1.11256
\(796\) 0 0
\(797\) 0.384472 0.0136187 0.00680935 0.999977i \(-0.497833\pi\)
0.00680935 + 0.999977i \(0.497833\pi\)
\(798\) 0 0
\(799\) 32.3153 1.14323
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −17.1619 −0.605631
\(804\) 0 0
\(805\) 5.75379 0.202794
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −45.0540 −1.58401 −0.792007 0.610512i \(-0.790964\pi\)
−0.792007 + 0.610512i \(0.790964\pi\)
\(810\) 0 0
\(811\) 26.8769 0.943775 0.471888 0.881659i \(-0.343573\pi\)
0.471888 + 0.881659i \(0.343573\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) −36.4924 −1.27827
\(816\) 0 0
\(817\) 2.56155 0.0896174
\(818\) 0 0
\(819\) 13.1231 0.458558
\(820\) 0 0
\(821\) −39.0540 −1.36299 −0.681497 0.731821i \(-0.738671\pi\)
−0.681497 + 0.731821i \(0.738671\pi\)
\(822\) 0 0
\(823\) −13.3002 −0.463615 −0.231808 0.972762i \(-0.574464\pi\)
−0.231808 + 0.972762i \(0.574464\pi\)
\(824\) 0 0
\(825\) 2.24621 0.0782030
\(826\) 0 0
\(827\) −34.7386 −1.20798 −0.603990 0.796992i \(-0.706423\pi\)
−0.603990 + 0.796992i \(0.706423\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 6.31534 0.219077
\(832\) 0 0
\(833\) 2.49242 0.0863573
\(834\) 0 0
\(835\) 43.8617 1.51790
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) −7.86174 −0.271417 −0.135709 0.990749i \(-0.543331\pi\)
−0.135709 + 0.990749i \(0.543331\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) 0 0
\(843\) 7.75379 0.267055
\(844\) 0 0
\(845\) 33.9309 1.16726
\(846\) 0 0
\(847\) 22.8769 0.786059
\(848\) 0 0
\(849\) 16.3153 0.559941
\(850\) 0 0
\(851\) 7.01515 0.240476
\(852\) 0 0
\(853\) −56.7386 −1.94269 −0.971347 0.237666i \(-0.923618\pi\)
−0.971347 + 0.237666i \(0.923618\pi\)
\(854\) 0 0
\(855\) −2.56155 −0.0876033
\(856\) 0 0
\(857\) −41.8617 −1.42997 −0.714985 0.699140i \(-0.753566\pi\)
−0.714985 + 0.699140i \(0.753566\pi\)
\(858\) 0 0
\(859\) −25.3002 −0.863231 −0.431616 0.902058i \(-0.642056\pi\)
−0.431616 + 0.902058i \(0.642056\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 0 0
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) 31.3693 1.06659
\(866\) 0 0
\(867\) −15.3153 −0.520136
\(868\) 0 0
\(869\) −19.2311 −0.652369
\(870\) 0 0
\(871\) −52.4924 −1.77864
\(872\) 0 0
\(873\) −12.2462 −0.414471
\(874\) 0 0
\(875\) 22.5616 0.762720
\(876\) 0 0
\(877\) −47.3693 −1.59955 −0.799774 0.600301i \(-0.795047\pi\)
−0.799774 + 0.600301i \(0.795047\pi\)
\(878\) 0 0
\(879\) −7.12311 −0.240256
\(880\) 0 0
\(881\) 23.3002 0.785003 0.392502 0.919751i \(-0.371610\pi\)
0.392502 + 0.919751i \(0.371610\pi\)
\(882\) 0 0
\(883\) 47.6847 1.60472 0.802358 0.596843i \(-0.203578\pi\)
0.802358 + 0.596843i \(0.203578\pi\)
\(884\) 0 0
\(885\) 30.7386 1.03327
\(886\) 0 0
\(887\) 31.2311 1.04864 0.524318 0.851522i \(-0.324320\pi\)
0.524318 + 0.851522i \(0.324320\pi\)
\(888\) 0 0
\(889\) −46.1080 −1.54641
\(890\) 0 0
\(891\) −1.43845 −0.0481898
\(892\) 0 0
\(893\) 5.68466 0.190230
\(894\) 0 0
\(895\) 17.6155 0.588822
\(896\) 0 0
\(897\) −4.49242 −0.149998
\(898\) 0 0
\(899\) 16.4924 0.550053
\(900\) 0 0
\(901\) 69.6155 2.31923
\(902\) 0 0
\(903\) −6.56155 −0.218355
\(904\) 0 0
\(905\) 17.6155 0.585560
\(906\) 0 0
\(907\) 34.8769 1.15807 0.579034 0.815303i \(-0.303430\pi\)
0.579034 + 0.815303i \(0.303430\pi\)
\(908\) 0 0
\(909\) −1.12311 −0.0372511
\(910\) 0 0
\(911\) −1.12311 −0.0372101 −0.0186051 0.999827i \(-0.505923\pi\)
−0.0186051 + 0.999827i \(0.505923\pi\)
\(912\) 0 0
\(913\) 5.75379 0.190423
\(914\) 0 0
\(915\) 14.5616 0.481390
\(916\) 0 0
\(917\) 21.3002 0.703394
\(918\) 0 0
\(919\) −44.4924 −1.46767 −0.733835 0.679328i \(-0.762271\pi\)
−0.733835 + 0.679328i \(0.762271\pi\)
\(920\) 0 0
\(921\) 28.4924 0.938857
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −12.4924 −0.410748
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 0.438447 0.0143695
\(932\) 0 0
\(933\) 7.93087 0.259645
\(934\) 0 0
\(935\) 20.9460 0.685008
\(936\) 0 0
\(937\) −16.5616 −0.541042 −0.270521 0.962714i \(-0.587196\pi\)
−0.270521 + 0.962714i \(0.587196\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) −2.73863 −0.0891822
\(944\) 0 0
\(945\) 6.56155 0.213447
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) −61.1231 −1.98414
\(950\) 0 0
\(951\) 27.1231 0.879527
\(952\) 0 0
\(953\) −11.6155 −0.376264 −0.188132 0.982144i \(-0.560243\pi\)
−0.188132 + 0.982144i \(0.560243\pi\)
\(954\) 0 0
\(955\) 35.0540 1.13432
\(956\) 0 0
\(957\) 11.8617 0.383436
\(958\) 0 0
\(959\) −17.4384 −0.563117
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −11.3693 −0.366371
\(964\) 0 0
\(965\) 69.4773 2.23655
\(966\) 0 0
\(967\) 26.7386 0.859856 0.429928 0.902863i \(-0.358539\pi\)
0.429928 + 0.902863i \(0.358539\pi\)
\(968\) 0 0
\(969\) −5.68466 −0.182618
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 32.8078 1.05177
\(974\) 0 0
\(975\) 8.00000 0.256205
\(976\) 0 0
\(977\) 6.98485 0.223465 0.111732 0.993738i \(-0.464360\pi\)
0.111732 + 0.993738i \(0.464360\pi\)
\(978\) 0 0
\(979\) 8.63068 0.275838
\(980\) 0 0
\(981\) 14.2462 0.454847
\(982\) 0 0
\(983\) −30.7386 −0.980410 −0.490205 0.871607i \(-0.663078\pi\)
−0.490205 + 0.871607i \(0.663078\pi\)
\(984\) 0 0
\(985\) 59.8617 1.90735
\(986\) 0 0
\(987\) −14.5616 −0.463499
\(988\) 0 0
\(989\) 2.24621 0.0714254
\(990\) 0 0
\(991\) 21.8617 0.694461 0.347231 0.937780i \(-0.387122\pi\)
0.347231 + 0.937780i \(0.387122\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 66.4233 2.10576
\(996\) 0 0
\(997\) −55.7926 −1.76697 −0.883485 0.468460i \(-0.844809\pi\)
−0.883485 + 0.468460i \(0.844809\pi\)
\(998\) 0 0
\(999\) 8.00000 0.253109
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 912.2.a.m.1.2 2
3.2 odd 2 2736.2.a.z.1.1 2
4.3 odd 2 456.2.a.f.1.2 2
8.3 odd 2 3648.2.a.bl.1.1 2
8.5 even 2 3648.2.a.br.1.1 2
12.11 even 2 1368.2.a.k.1.1 2
76.75 even 2 8664.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.a.f.1.2 2 4.3 odd 2
912.2.a.m.1.2 2 1.1 even 1 trivial
1368.2.a.k.1.1 2 12.11 even 2
2736.2.a.z.1.1 2 3.2 odd 2
3648.2.a.bl.1.1 2 8.3 odd 2
3648.2.a.br.1.1 2 8.5 even 2
8664.2.a.r.1.2 2 76.75 even 2